United and City are both interested in Kane, whose Tottenham future is up in the air this summer amid reports he has informed Spurs bosses he would like to leave.
City boss Pep Guardiola and United manager Ole Gunnar Solskjaer could both benefit from a striker signing and Kane is an elite forward who could be available.
Sergio Aguero’s exit has left a gap open in City’s striker department, while Edinson Cavani is only a short-term solution at Old Trafford.
Kane claimed the Golden Boot and Playmaker of the Season award in the Premier League despite Spurs finishing in a lowly seventh.
The 27-year-old netted 23 goals and contributed another 14 assists for his teammates in the league.
The England captain shared a good relationship with former boss Mourinho before the Portuguese tactician’s sacking in April.
New Roma boss Mourinho has noted Kane’s leadership qualities for potential suitors but suggested he could also benefit from captaincy at his club.
“He is a leader – a silent leader, but a leader,” he told the Times.
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“In the club it is not an easy situation for a leader who does not have the armband. It creates a little bit of uncertainty about the roles.
“In the national team he has that armband that gives him, let’s say, the institutional power to be a leader of the group.”
Kane will lead England at the Euros still without a trophy after another season ended without lifting a major honour at Spurs.
His best chance in the last campaign came in the League Cup final defeat to City at Wembley, which Mourinho was sacked just days prior to, despite his speciality in finals.
Kane was twice involved in a title race under Mauricio Pochettino’s reign and the Argentine boss also led the North London club to the Champions League final.
But the lack of silverware is likely to be the ultimate reason Kane would seek a new challenge after the European Championships.
And Mourinho believes Kane’s trophyless campaign will make him even more hungry for silverware with England this summer.
“Harry [Kane] is the one that has on his shoulders a lot,” Mourinho said, speaking to The Sun.
“The country expects a lot from him. He expects a lot from himself.
“He’s the captain and he wants to win something. He’s not doing that with the club, so, of course, he’s going to try everything he can to do it with the Three Lions.”